The theory of partial differential equations constitutes today one of the most important topics of scientific understanding. A standard approach for solving a time-dependent partial differential equation consists in discretizing the spatial variables by finite differences or finite elements. This results in a huge system of (stiff) ordinary differential equations that has to be integrated in time. Exponential integrators constitute an interesting class of numerical methods for the time integration of stiff systems of differential equations. Their efficient implementation heavily relies on the fast computation of the action of certain matrix functions; among those, the matrix exponential is the most prominent one. In this manuscript, we go through the steps that led to the development of backward error accurate routines for computing the action of the matrix exponential.

This thesis gives an overview of the process that led to the development of a novel semi-implicit fluid-structure interaction model. The thesis is dedicated to the creation of a new numerical model that allows to study ship generated waves and ship manoeuvers in waterways for various vessel characteristics and speeds in different external current situations. A model like this requires a coupling between the fluid and the solid to generate the waves and the hydrodynamic forces on the hull. Since the horizontal dimensions are significantly larger than the vertical dimension, we started by employing the shallow water equations, which are based on the assumption of hydrostatic pressure. The discretization was carried out taking only the nonlinear advective terms explicitly while the pressure terms are discretized implicitly, which makes the CFL condition milder. The price to pay for this semi-implicit discretization is an increase in the algorithm complexity compared to a fully-explicit method, but it is still much simpler than a fully-implicit discretization of the governing equations. Indeed, the mass and momentum equations couple, and finding the unknowns involves solving a system of equations with dimensions equal to the number of cells. The grid supporting the discretization is staggered, overlapping and Cartesian. Since the aimed application domain is inland waterways, it is paramount to allow wetting and drying of the cells. This was achieved by acting on the depth function, the relationship between the free-surface elevation and the water depth in the cell. The main novelty of this research project is the two-way coupling of the PDE system for the water flow with the ODE system for the rigid body motion of the ship. The hull defines the ship region, and its shape can range from a simple box to an STL file of a real 3D ship geometry. Where the hull is in contact with the water, the cells are pressurized. This pressurized group of cells generates waves as it moves, and its motion is influenced by incoming external waves. This result is obtained by imposing an upper bound to the depth function, so that the water depth does not increase when it reaches the hull elevation, while the pressure is allowed to increase. This upper bound increases the nonlinearity of the system, which may have dry cells, wet free-surface cells and pressurized cells. The solution of this system is found by a single nested-Newton iterative solver of Casulli and Zanolli [36], in which with two separate linearizations the system is written in a sparse, symmetric, positive semi-definite form. This particular form allows us to employ a matrix-free conjugate gradient method, and efficiently get the unknown pressure. The integral of the pressure over the hull is applied for the hydrodynamic force and torque acting on the ship. After adding the skin friction and other external forces from the propeller or the rudder, the total force is inserted in the equation of motion of the rigid body. The ODE system is discretized with a second-order Taylor method, and it is solved for the six degrees of freedom (3 coordinates for the position vector of the barycenter and 3 rotation angles), providing the next position and orientation of the ship. The vertical translation of the rigid body is governed by the gravitational force and the restoring force from Archimedes' principle. As the ship oscillates up and down, the gravitational potential energy is partially transferred to the radiated free-surface water waves, damping and eventually stopping the motion. Also, the ship pushes and pulls the water around it, inducing the added mass force. All these elements constitute the ODE that was used for the verification of the vertical degree of freedom. The numerical simulation gave the expected results for the vertical motion. The horizontal translation, important for the manoeuvers, presented a numerical instability unseen in our previous test cases, which is connected to the relative motion between the ship and the grid. In each time step in which the ship enters a new cell, the pressure sharply increases and decreases at the ship bow. An oscillation can build up in time and create an unphysical void below the vessel. We implemented a few ideas to attenuate the oscillations. At the heart of all the following techniques is the reduction of the time derivative of the water depth, especially for those cells transitioning to a pressurized state. All these modifications were effective at controlling the oscillations, each with a different intensity, and simulations with a horizontal motion are much more stable than without these techniques. With the collaboration of the BAW research institute, we worked on the model validation. We used data from two separate experiments to compare the measurements with the numerical results. Specifically, we focused on the ship-generated wave height and the hydrodynamic forces on the hull. The comparison is satisfactory for the wave height. The force and torque prediction is plausible but underestimated compared to the measurements. The model seems to displace the water volume correctly during the ship passage, while the force and torque response might need additional work to be trusted in applications. Even though the hydrostatic assumption is mostly correct in our range of applications, the presence and the motion of a ship could generate strong vertical accelerations of the flow, which may not be negligible. For this reason, we implemented an algorithm that corrects the velocity field, introducing also dispersive effects due to a non-hydrostatic pressure. The correction consists of a higher-order Bousinnesq-type term in the momentum equation and the solution of the resulting system. The non-hydrostatic update has a small influence on the wave generation, while it alters significantly the reaction forces. The subgrid method implementation allowed to benefit from high-resolution bottom descriptions while keeping the grid size coarse. The same subgrid can also be used for a refined definition of the hull, which makes the volume computations more accurate. Furthermore, the subgrid introduces new possible states for the cells, as they can be partially dry or partially pressurized. These intermediate states translate into smoother transitions from one state to the other when the free-surface is close to the bathymetry or to the hull. Concerning the software implementation of the developed scheme, in order to improve the execution performance of the prototype script formulated initially in Matlab, the numerical method was rewritten as a Fortran program. Also, thanks to the domain decomposition technique and the MPI standard, each simulation can run in parallel on multiple CPUs, leveraging the computational power of supercomputers. The coupling of the PDE and ODE system, together with an appropriate redefinition of the depth function, proved to be a valuable method for studying fluid-structure interaction problems. The combination of efficient numerical techniques led to the development of a tool with a potential to be applied in the practice for the simulation of floating objects in wide domains.

The present work analyzes and describes a method for the direct numerical solution of the Maxwell's equations of classical electromagnetism. This is the FDTD (Finite-Difference Time-Domain) method, along with its implementation in an "in-house" computing code for large parallelized simulations. Both are then applied to the modelization of photonic and plasmonic structures interacting with light. These systems are often too complex, either geometrically and materially, in order to be mathematically tractable and an exact analytic solution in closed form, or as a series expansion, cannot be obtained. The only way to gain insight on their physical behavior is thus to try to get a numerical approximated, although convergent, solution. This is a current trend in modern physics because, apart from perturbative methods and asymptotic analysis, which represent, where applicable, the typical instruments to deal with complex physico-mathematical problems, the only general way to approach such problems is based on the direct approximated numerical solution of the governing equations. Today this last choice is made possible through the enormous and widespread computational capabilities offered by modern computers, in particular High Performance Computing (HPC) done using parallel machines with a large number of CPUs working concurrently. Computer simulations are now a sort of virtual laboratories, which can be rapidly and costless setup to investigate various physical phenomena. Thus computational physics has become a sort of third way between the experimental and theoretical branches. The plasmonics application of the present work concerns the scattering and absorption analysis from single and arrayed metal nanoparticles, when surface plasmons are excited by an impinging beam of light, to study the radiation distribution inside a silicon substrate behind them. This has potential applications in improving the eciency of photovoltaic cells. The photonics application of the present work concerns the analysis of the optical reflectance and transmittance properties of an opal crystal. This is a regular and ordered lattice of macroscopic particles which can stops light propagation in certain wavelenght bands, and whose study has potential applications in the realization of low threshold laser, optical waveguides and sensors. For these latters, in fact, the crystal response is tuned to its structure parameters and symmetry and varies by varying them. The present work about the FDTD method represents an enhacement of a previous one made for my MSc Degree Thesis in Physics, which has also now geared toward the visible and neighboring parts of the electromagnetic spectrum. It is organized in the following fashion. Part I provides an exposition of the basic concepts of electromagnetism which constitute the minimum, although partial, theoretical background useful to formulate the physics of the systems here analyzed or to be analyzed in possible further developments of the work. It summarizes Maxwell's equations in matter and the time domain description of temporally dispersive media. It addresses also the plane wave representation of an electromagnetic field distribution, mainly the far field one. The Kirchhoff formula is described and deduced, to calculate the angular radiation distribution around a scatterer. Gaussian beams in the paraxial approximation are also slightly treated, along with their focalization by means of an approximated diraction formula useful for their numericall FDTD representation. Finally, a thorough description of planarly multilayered media is included, which can play an important ancillary role in the homogenization procedure of a photonic crystal, as described in Part III, but also in other optical analyses. Part II properly concerns the FDTD numerical method description and implementation. Various aspects of the method are treated which globally contribute to a working and robust overall algorithm. Particular emphasis is given to those arguments representing an enhancement of previous work.These are: the analysis from existing literature of a new class of absorbing boundary conditions, the so called Convolutional-Perfectly Matched Layer, and their implementation; the analysis from existing literature and implementation of the Auxiliary Differential Equation Method for the inclusion of frequency dependent electric permittivity media, according to various and general polarization models; the description and implementation of a "plane wave injector" for representing impinging beam of lights propagating in an arbitrary direction, and which can be used to represent, by superposition, focalized beams; the parallelization of the FDTD numerical method by means of the Message Passing Interface (MPI) which, by using the here proposed, suitable, user dened MPI data structures, results in a robust and scalable code, running on massively parallel High Performance Computing Machines like the IBM/BlueGeneQ with a core number of order 2X10^5. Finally, Part III gives the details of the specific plasmonics and photonics applications made with the "in-house" developed FDTD algorithm, to demonstrate its effectiveness. After Chapter 10, devoted to the validation of the FDTD code implementation against a known solution, Chapter 11 is about plasmonics, with the analytical and numerical study of single and arrayed metal nanoparticles of different shapes and sizes, when surface plasmon are excited on them by a light beam. The presence of a passivating embedding silica layer and a silicon substrate are also included. The next Chapter 12 is about the FDTD modelization of a face-cubic centered (FCC) opal photonic crystal sample, with a comparison between the numerical and experimental transmittance/reflectance behavior. An homogenization procedure is suggested of the lattice discontinuous crystal structure, by means of an averaging procedure and a planarly multilayered media analysis, through which better understand the reflecting characteristic of the crystal sample. Finally, a procedure for the numerical reconstruction of the crystal dispersion banded omega-k curve inside the first Brillouin zone is proposed. Three Appendices providing details about specific arguments dealt with during the exposition conclude the work.

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as φ-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed.

In the present PhD thesis an optimal problem suite is proposed as benchmark for the test of numerical solvers. The problems are divided in four categories, classic, singular, constrained and hard problems. Apart from the hard problems, where it is not possible to give the analytical solution but only some details, all other problems are supplied with the derivation of the solution. The exact solution allows a precise comparison of the performance of the considered software. All of the proposed problems were taken from published papers or books, but it turned out that an analytic exact solution was only rarely provided, thus a true and reliable comparison among numerical solvers could not be done before. A typical wrong conclusion when a solver obtains a lower value of the target functional with respect to other solvers is to claim it better than the others, but it is not recognized that it has only underestimated the true value. In this thesis, a cutting edge application of optimal control to vehicles is showed: the optimization of the lap time in a race circuit track considering a number of realistic constraints. A new algorithm for path planning is completely described for the construction of a quasi G2 fitting of the GPS data with a clothoid spline in terms of the G1 Hermite interpolation problem. In particular the present algorithm is proved to work better than state of the art algorithms in terms of both efficiency and precision.

It is currently a well-established fact that the dynamics of interacting fluid compartments of the central nervous system (CNS) may play a role in the CNS fluid physiology and pathology of a number of neurological disorders, including neurodegenerative diseases associated with accumulation of waste products in the brain. However, the mechanisms and routes of waste clearance from the brain are still unclear. One of the main components of this interacting cerebral fluids dynamics is blood flow. In the last decades, mathematical modeling and fluid dynamics simulations have become a valuable complementary tool to experimental approaches, contributing to a deeper understanding of the circulatory physiology and pathology. However, modeling blood flow in the brain remains a challenging and demanding task, due to the high complexity of cerebral vascular networks and the difficulties that consequently arise to describe and reproduce the blood flow dynamics in these vascular districts. The first part of this work is devoted to the development of efficient numerical strategies for blood flow simulations in complex vascular networks. In cardiovascular modeling, one-dimensional (1D) and lumped-parameter (0D) models of blood flow are nowadays well-established tools to predict flow patterns, pressure wave propagation and average velocities in vascular networks, with a good balance between accuracy and computational cost. Still, the purely 1D modeling of blood flow in complex and large networks can result in computationally expensive simulations, posing the need for extremely efficient numerical methods and solvers. To address these issues, we develop a novel modeling and computational framework to construct hybrid networks of coupled 1D and 0D vessels and to perform computationally efficient and accurate blood flow simulations in such networks. Starting from a 1D model and a family of nonlinear 0D models for blood flow, with either elastic or viscoelastic tube laws, this methodology is based on (i) suitable coupling equations ensuring conservation principles; (ii) efficient numerical methods and numerical coupling strategies to solve 1D, 0D and hybrid junctions of vessels; (iii) model selection criteria to construct hybrid networks, which provide a good trade-off between accuracy in the predicted results and computational cost of the simulations. By applying the proposed hybrid network solver to very complex and large vascular networks, we show how this methodology becomes crucial to gain computational efficiency when solving networks and models where the heterogeneity of spatial and/or temporal scales is relevant, still ensuring a good level of accuracy in the predicted results. Hence, the proposed hybrid network methodology represents a first step towards a high-performance modeling and computational framework to solve highly complex networks of 1D-0D vessels, where the complexity does not only depend on the anatomical detail by which a network is described, but also on the level at which physiological mechanisms and mechanical characteristics of the cardiovascular system are modeled. Then, in the second part of the thesis, we focus on the modeling and simulation of cerebral blood flow, with emphasis on the venous side. We develop a methodology that, departing from the high-resolution MRI data obtained from a novel in-vivo microvascular imaging technique of the human brain, allows to reconstruct detailed subject-specific cerebral networks of specific vascular districts which are suitable to perform blood flow simulations. First, we extract segmentations of cerebral districts of interest in a way that the arterio-venous separation is addressed and the continuity and connectivity of the vascular structures is ensured. Equipped with these segmentations, we propose an algorithm to extract a network of vessels suitable and good enough, i.e. with the necessary properties, to perform blood flow simulations. Here, we focus on the reconstruction of detailed venous vascular networks, given that the anatomy and patho-physiology of the venous circulation is of great interest from both clinical and modeling points of view. Then, after calibration and parametrization of the MRI-reconstructed venous networks, blood flow simulations are performed to validate the proposed methodology and assess the ability of such networks to predict physiologically reasonable results in the corresponding vascular territories. From the results obtained we conclude that this work represents a proof-of-concept study that demonstrates that it is possible to extract subject-specific cerebral networks from the novel high-resolution MRI data employed, setting the basis towards the definition of an effective processing pipeline for detailed blood flow simulations from subject-specific data, to explore and quantify cerebral blood flow dynamics, with focus on venous blood drainage.

The main bulk of this Thesis is focused on the response of cell membranes due to chemical and mechanical stimuli. Henceforth, it is mainly devoted to deduce how the key aspect of the cell response activated by chemical signaling can be predicted by a simplified energetics, making use of both theoretical models and numerical simulations. The a ention is focused on cell membranes embedding G protein-coupled receptors (GPRCs). By analyzing the behavior of cell mem- branes, one can isolate three main contributions in order to model their respon- se: (1) diffusion of receptors and transporters embedded in the lipid membrane; (2) conformational changes of the receptors; (3) membrane elasticity. Moreover, the interplay between TM confomational changes and lateral pressure of the lipid membrane against such TMs is introduced. The chemical potential of the receptor-ligand compound, deduced as the variational derivative of such energy, is compared with the one calculated by accounting for the work done by the lateral pressure. The result yields a relationship between the conformational field, the mechanical field (interpreted as either the thickness change or the areal change) and the distribution of the compounds receptor-ligand. The analysis of such resulting constitutive equation among those three quantities shows that, essentially, the reason why ligand-GPRCs compounds prefer to live on lipid ra is a necessity involving the interplay between the work performed by the lateral pressure and the need of TMs to change their conformation during ligand binding. Henceforth, mechanobiology gives a justification to the experimental findings of Kobilka and Lei ovitz, Chemistry Nobel Prizes 2012.

In this Thesis we explore, both theoretically and numerically, splitting strategies for a hyperbolic system of one-dimensional (1D) blood flow equations with a passive scalar transport equation. Our analysis involves a two-step framework that includes splitting at the level of partial differential equations (PDEs) and numerical methods for discretizing the ensuing problems. This study is inspired by the original flux splitting approach of Toro and Vázquez-Cendón (2012) originally developed for the conservative Euler equations of compressible gas dynamics. In this approach the flux vector in the conservative case, and the system matrix in the non-conservative one, are split into advection and pressure terms: in this way, two systems of partial differential equations are obtained, the advection system and the pressure system. From the mathematical as well as numerical point of view, a basic problem to be solved is the special Cauchy problem called the Riemann problem. This latter provides an analytical solution to evaluate the performance of the numerical methods and, in our approach, it is of primary importance to build the presented numerical schemes. In the first part of the Thesis a detailed theoretical analysis is presented, involving the exact solution of the Riemann problem for the 1D blood flow equations, depicted for a general constant momentum correction coefficient and a tube law that allows to describe both arteries and veins with continuous or discontinuous mechanical and geometrical properties and an advection equation for a passive scalar transport. In literature, this topic has been already studied only for a momentum correction coefficient equal to one, that is related to the prescribed velocity profile and in this case corresponds to a flat one, i.e. an inviscid fluid. In the case of discontinuous properties, only the subsonic regime is considered. In addition we propose a procedure to compute the obtained exact solution and finally we validate it numerically, by comparing exact solutions to those obtained with well-known, numerical schemes on a carefully designed set of test problems. Furthermore, an analogous theoretical analysis and resolution algorithm are presented for the advection system and the pressure system arising from the splitting at the level of PDEs of the complete system of 1D blood flow equations. It is worth noting that the pressure system, in case of veins, presents a loss of genuine non-linearity resulting in the formation of rarefactions, shocks and compound waves, these latter being a composition of rarefactions and shocks. In the second part of the Thesis we present novel finite volume-type, flux splitting-based, numerical schemes for the conservative 1D blood flow equations and splitting-based numerical schemes for the non-conservative 1D blood flow equations that incorporate an advection equation for a passive scalar transport, considering tube laws that allow to model blood flow in arteries and veins and take into account a general constant momentum correction coefficient. A detailed efficiency analysis is performed in order to showcase the advantages of the proposed methodologies in comparison to standard approaches.

In modern engineering, the accurate and efficient numerical simulation of dynamic systems is crucial, providing valuable insights across various fields such as automotive, aerospace, robotics, and electrical engineering. These simulations help to understand system behaviors, to optimize performance, and to guide design decisions. Nonetheless, systems described by Ordinary Differential Equations (ODEs) and Differential-Algebraic Equations (DAEs) are central to such simulations. While ODEs can be easily solved, they often fall short of modeling systems with constraints or algebraic relationships. DAEs, however, offer a more comprehensive framework, making them suitable for a wider range of dynamic systems. However, the inherent complexity of DAEs poses significant challenges for numerical integration and solution. In vehicle dynamics, the simulation of systems described by DAEs is particularly relevant. The advances in autonomous and high-performance cars rely heavily on robust simulations that accurately reflect the interactions between mechanical components, control systems, and environmental factors. Achieving accuracy and speed in these simulations is critical for Real-Time (RT) applications, where rapid decision-making and control are essential. The challenges faced in vehicle dynamics simulations, such as equations' stiffness and complexity, are representative of broader issues in dynamic system simulations. This thesis addresses these challenges by integrating symbolic computation with numerical methods to solve DAEs efficiently and accurately. Specifically, the index reduction approach transforms high-index DAEs into low-index formulations more suitable for numerical integration, enhancing the speed and stability of solvers. Symbolic computation, which handles mathematical expressions in their exact form, aids this process by simplifying the involved expressions, detecting redundancies and symbolic cancellations, and thereby ensuring equations' consistency while keeping complexity at the minimum. Hence, combining symbolic and numerical methods leverages the strengths of both techniques, aiming at improved performance and reliability. Such a hybrid framework is designed to handle the specific requirements of vehicle dynamics and other applications in engineering. The thesis encompasses several advancements in dynamic system simulation by integrating symbolic computation with numerical methods to reduce computational overhead and improve performance. The research focuses on developing new algorithms for DAEs index reduction, transforming high-index DAEs into more suitable for standard numerical integration methods. Specifically, such an index reduction process is based on symbolic matrix factorization with simultaneous expression swell mitigation. This novel methodology is validated through practical applications, applying the proposed technique to real-world simulation problems to assess its performance, accuracy, and efficiency. Additionally, the research aims to enhance Hard Real-Time (HRT) vehicle dynamic simulation by designing dedicated algorithms and models for simulating tire-road interactions and vehicle structures' deformation, improving both speed and fidelity. Altogether, this thesis introduces several open-source software libraries made available to the research community with comprehensive documentation and examples. In summary, this work bridges the gap between symbolic computation and numerical methods for the simulation of dynamic systems described by DAEs. Thanks to mixed symbolic-numeric frameworks, innovative algorithms, and practical tools, it contributes to the advancement of simulation techniques, setting the stage for further investigations and applications in engineering.

Very few standards exist for tting products to people. Footwear fit is a noteworthy example for consumer consideration when purchasing shoes. As a result, footwear manufacturing industry for achieving commercial success encountered the problem of developing right footwear which is fulfills consumer's requirement better than it's competeries. Mass customization starts with understanding individual customer's requirement and it finishes with fulllment process of satisfying the target customer with near mass production efficiency. Unlike any other consumer product, personalized footwear or the matching of footwear to feet is not easy if delivery of discomfort is predominantly caused by pressure induced by a shoe that has a design unsuitable for that particular shape of foot. Footwear fitter have been using manual measurement for a long time, but the combination of 3D scanning systems with mathematical technique makes possible the development of systems, which can help in the selection of good footwear for a given customer. This thesis, provides new approach for addressing the computerize footwear fit customization in industry problem. The design of new shoes starts with the design of the new shoe last. A shoe last is a wooden or metal model of human foot on which shoes are shaped. Despite the steady increase in accuracy, most available scanning techniques cause some deficiencies in the point cloud and a set of holes in the triangle meshes. Moreover, data resulting from 3D scanning are given in an arbitrary position and orientation in a 3D space. To apply sophisticated modeling operations on these data sets, substantial post-processing is usually required. We described a robust algorithm for filling holes in triangle mesh. First, the advance front mesh technique is used to generate a new triangular mesh to cover the hole. Next, the triangles in initial patch mesh is modified by estimating desirable normals instead of relocating them directly. Finally, the Poisson equation is applied to optimize the new mesh. After obtaining complete 3D model, the result data must be generated and aligned before taking this models for shape analysis such as measuring similarity between foot and shoe last data base for evaluating footwear it. Principle Component Analysis (PCA), aligns a model by considering its center of mass as the coordinate system origin, and its principle axes as the coordinate axes. The purpose of the PCA applied to a 3D model is to make the resulting shape independent to translation and rotation asmuch as possible. In analysis, we applied "weighted" PCA instead of applying the PCA in a classical way (sets of 3D point-clouds) for alignment of 3D models. This approach is based on establishing weights associated to center of gravity of triangles. When all of the models are aligned, an efficient algorithm to cut the model to several sections toward the heel and toe for extracting counters is used. Then the area of each contour is calculated and compared with equal sections in shoe last data base for finding best footwear fit within the shoe last data base.